You won’t get more energy out than the battery contains. The more changes you put the energy through the more losses you get, mostly due to entropy, the rest through inefficiencies.
When you burn H2-O2 you are still constrained by Carnot thermal efficiency.
If he can go 300 miles by electrolysing then burning water, he’d go much furhter on just the battery.
The bike will have an alternator recharging the battery. It will need to in order to fire the spark plugs and burn the gasoline from the hidden tank in the black case.